The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

This paper carries out an investigation of the orthogonal-polynomial approach to reshaping symmetric distributions to fit in with data requirements so as to cover the multivariate case. With this objective in mind, reference is made to the class of spherical distributions, given that they provide a natural multivariate generalization of univariate even densities. After showing how to tailor a spherical distribution via orthogonal polynomials to better comply with kurtosis requirements, we provide operational conditions for positiveness of the resulting multivariate Gram-Charlier-like expansion together with its kurtosis range. The approach here proposed is finally applied to some selected spherical distributions.